This invention relates to the deconvolution of seismic data and specifically to the determination and application of filters to seismic traces and stacked data and the generation of graphic displays and representations from the data.
Deconvolution is a process that is applied to seismic data to diminish the distortion effects of the seismic wavelet as the seismic signal travels through, and is reflected from, the subsurface strata, in order to recover the reflection coefficients from the seismic trace. Conventional deconvolution methods assume that the reflection coefficients have a white noise behavior, i.e., that they are uncorrelated random variables, with a flat power spectrum having a slope of zero and a spike auto-correlation function. A detailed discussion and description of conventional white noise deconvolution as practiced in the prior art is provided by E. A. Robinson and S. Treitel in Principles of Digital Weiner Filtering, Geophysical Prospecting, 15, page 311-333 (1967) and Geophysical Signal Analysis, Prentice-Hall (1980).
The assumption that the reflection coefficients have a random white noise behavior has been necessary as a practical matter in order to facilitate computations on the raw data, even though the reflection coefficients are observed in nature to depart from that model. However, this assumption of white noise behavior adversely affects the accuracy of the deconvolution process and use of that model limits analysis of the data recovered and displayed for interpretation to only the White component of reflectivity.
Various methods have been proposed and employed by the prior art for processing seismic data employing filters in an effort to improve the representations of the processed data. U.S. Pat. No. 3,689,874 discloses a seismic processing scheme employing conventional inverse filtering and smoothing techniques. U.S. Pat. No. 4,630,242 discloses two methods for estimating the earth""s reflectivity sequence, the first based on Kalman filtering and the second on Weiner filtering and pilot auto-correlation data. U.S. Pat. No. 5,010,525 discloses a method for filtering noise bands from the data and U.S. Pat. No. 5,237,538 describes a method for filtering coherent noise bands from seismic a data. U.S. Pat. No. 4,884,247 describes a method that is essentially Q-filtering that compensates for the effects of attenuation on the wavelet""s passage through the earth.
Numerous other methods have been proposed for improving the identification of stratigraphic prospects including the removal of sinusoidal noise from seismic data, U.S. Pat. Nos. 4,853903 and 5,051,963; the use of ceptral windows for deconvolution of the signal amplitude spectrum, U.S. Pat. No. 4,780,859; and the use of correlating signals to combine seismic data that have different spectral characteristics, U.S. Pat. No. 4,715,021.
Although some of these prior art methods have been adopted for the processing of seismic data to produce graphic representations and displays of the earth""s strata, the resulting displays lack precision and completeness.
It is therefore a principal object of the invention to provide an improved method for the processing and analysis of seismic data that more accurately models the earth""s reflectivity and performs a generalized form of deconvolution that takes into account the correct stochastic behavior of reflectivity.
Another principal object of the invention is to provide a method for the deconvolution of seismic data that is based on a fractionally integrated noise model of the earth""s reflectivity.
Another object of the invention is to provide a method of processing seismic data to produce improved graphic displays of the seismic lines that represent the earth""s strata.
Yet another object of the invention is to provide a method of processing seismic data based on the non-random stochastic behavior of reflectivity that is more accurate than the white noise model of reflectivity.
It is a further object of this invention to provide a method of defining a filter for processing seismic data that yields significantly more accurate output data and associated graphic displays than was possible employing the conventional filters of the prior art.
It is another object of the invention to provide an improved method of deconvoluting seismic data that more accurately represents the effects on the seismic wavelets of multiple reflections and that can be applied to both traces and to stacks.
The above objects and other benefits and advantages are attained by the method of the invention that models the earth""s reflectivity by fractionally integrated noise and employs a generalized form of deconvolution that takes into account the correct stochastic behavior of reflectivity. The improved method yields significantly more accurate output, e.g., seismic line displays, than the conventional white noise filter of the prior art.
As used herein, the term xe2x80x9cFractionally Integrated Noisexe2x80x9d means a fractal stochastic process that results from fractionally integrating white noise. It has a single parameter d, the process order, which describes its correlation behavior. The auto-correlation function can be represented by the following:                                           ρ            y                    ⁡                      (            k            )                          =                              Γ            ⁢                          xe2x80x83                        ⁢                          (                              1                -                d                            )                        ⁢            Γ            ⁢                          xe2x80x83                        ⁢                          (                              k                +                d                            )                                                          Γ              ⁡                              (                d                )                                      ⁢                          Γ              ⁡                              (                                  k                  +                  1                  -                  d                                )                                                                        (        1        )            
where xcfx81y(k) is the auto-correlation function at lag k, d is the process order, and xcex93 is the Gamma function.
The power spectrum is represented by the following:                                           P            y                    ⁡                      (            f            )                          =                              σ            2                    ⁢                      π                    ⁢                                    Γ              ⁢                              xe2x80x83                            ⁢                              (                                  1                  -                  d                                )                                                    Γ              ⁢                              xe2x80x83                            ⁢                              (                                                      1                    2                                    -                  d                                )                                              ⁢                      xe2x80x83                    ⁢                                    sin                                                -                  2                                ⁢                d                                      ⁡                          (                              π                ⁢                                  xe2x80x83                                ⁢                f                            )                                                          (        2        )            
where Py is the power spectrum, "sgr"2 is the variance, the mean being assumed to be zero, and ƒ is the frequency normalized by the folding frequency (and hence, 0 less than ƒ less than xc2xd). The power spectrum of equation (2) is normalized so that the process has unit variance.
The method of the invention can be conveniently divided into the following three principal steps:
1. estimating the order of the fractionally integrated noise model;
2. computing one or both of the correction filters based on the fractionally integrated noise model; and
3. applying one or both of the filters to the deconvolution processing of the data.
Following deconvolution processing of the data it can be displayed on a monitor and/or printed in the customary graphical representation for interpretation and analysis of the seismic lines.
As will be understood by those of ordinary skill in the art, the data resulting from the application of the filter(s) in step 3, above, can be subjected to further processing, e.g., stacking, prior to it being displayed.
As indicated in the above description of the invention, two different correction filters are provided, each of which can be utilized independently of the other, which gives rise to the implementation of two embodiments of the invention. For the purpose of further describing the invention, these filters are referred to as the xe2x80x9cReflectivity Whitening Filterxe2x80x9d and the xe2x80x9cSpectral Compensation Filterxe2x80x9d, which as used herein, have the following meanings: xe2x80x9cReflectivity Whitening Filterxe2x80x9d is a filter that removes the non-white component of reflectivity from the trace, leaving only the white component in the trace, the function of this filter being to xe2x80x9cwhitenxe2x80x9d the reflectivity sequence by retaining only the white component of reflectivity; the xe2x80x9cSpectral Compensation Filterxe2x80x9d is a filter that corrects the reflectivity sequence by compensating for the distortion induced in the spectral density by the conventional deconvolution method.